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The first 4 derivatives are cos(x) then -sin(x) then -cos(x) and then sin(x)

sin(pi/4) + cos(pi/4) * (x-pi/4) -sin(pi/4) * (1/2!) (x-pi/4)^2 .....

is that it?

pretty straightforward.. what was your question about?

that's the first 3 terms... you should be able to get the 4th now?

That does not look right though

well... I lied a bit... a few of those terms are zero...

but you can work from them... right?

f(x) = f(x0) + (x-x0)*f'(x0)/1! + (x-x0)^2*f"(x0)/2! + (x-x0)^3*f'"(x0)/3!